For next Tuesday Read chapter 8 No written homework Initial posts - PowerPoint PPT Presentation

For next Tuesday • Read chapter 8 • No written homework • Initial posts due Thursday 1pm and responses due by class time Tuesday

Program 1 • Any questions?

Imperfect Knowledge • What issues arise when we don’t know everything (as in standard card games)?

State of the Art • Chess – Deep Blue, Hydra, Rybka • Checkers – Chinook (alpha-beta search) • Othello – Logistello • Backgammon – TD-Gammon (learning) • Go • Bridge • Scrabble

Games/Mainstream AI

What about the games we play?

Knowledge • Knowledge Base – Inference mechanism (domain-independent) – Information (domain-dependent) • Knowledge Representation Language – Sentences (which are not quite like English sentences) – The KRL determine what the agent can “know” – It also affects what kind of reasoning is possible • Tell and Ask

Getting Knowledge • We can TELL the agent everything it needs to know • We can create an agent that can “learn” new information to store in its knowledge base

The Wumpus World • Simple computer game • Good testbed for an agent • A world in which an agent with knowledge should be able to perform well • World has a single wumpus which cannot move, pits, and gold

Wumpus Percepts • The wumpus’s square and squares adjacent to it smell bad. • Squares adjacent to a pit are breezy. • When standing in a square with gold, the agent will perceive a glitter. • The agent can hear a scream when the wumpus dies from anywhere • The agent will perceive a bump if it walks into a wall. • The agent doesn’t know where it is.

Wumpus Actions • Go forward • Turn left • Turn right • Grab (picks up gold in that square) • Shoot (fires an arrow forward--only once) – If the wumpus is in front of the agent, it dies. • Climb (leave the cavern--only good at the start square)

Consequences • Entering a square containing a live wumpus is deadly • Entering a square containing a pit is deadly • Getting out of the cave with the gold is worth 1,000 points. • Getting killed costs 10,000 points • Each action costs 1 point

Possible Wumpus Environment Breeze Stench Pit Stench Breeze Breeze Stench Pit Wumpus Gold Stench Breeze Breeze Breeze Agent Pit

Knowledge Representation • Two sets of rules: – Syntax: determines what atomic symbols exist in the language and how to combine them into sentences – Semantics: Relationship between the sentences and “the world” --needed to determine truth or falsehood of the sentences

Reasoning • Entailment • Inference – May produce new sentences entailed by KB – May be used to determine which a particular sentence is entailed by the KB • We want inference procedures that are sound, or truth-preserving.

What Is a Logic? • A set of language rules – Syntax – Semantics • A proof theory – A set of rules for deducing the entailments of a set of sentences

Distinguishing Logics Language Ontological Epistemological Commitment (what Commitment (What exists in the world) an agent believes about facts) Propositional facts true/false/unknown Logic First-order logic facts, objects, true/false/unknown relations Temporal logic facts, objects, true/false/unknown relations, times degree of belief 0…1 Probability theory facts degree of belief 0…1 Fuzzy logic degree of truth

Propositional Logic • Simple logic • Deals only in facts • Provides a stepping stone into first order logic

Syntax • Logical Constants: true and false • Propositional symbols P, Q ... are sentences • If S is a sentence then (S) is a sentence. • If S is a sentence then ¬S is a sentence. • If S 1 and S 2 are sentences, then so are: – S 1  S 2 – S 1  S 2 – S 1  S 2 – S 1  S 2

Semantics • true and false mean truth or falsehood in the world • P is true if its proposition is true of the world • ¬S is the negation of S • The remainder follow standard truth tables – S 1  S 2 : AND – S 1  S 2 : inclusive OR – S 1  S 2 : True unless S 1 is true and S 2 is false – S 1  S 2 : bi-conditional, or if and only if

Vocabulary • An interpretation is an assignment of true or false to each atomic proposition • A sentence true under any interpretation is valid (a tautology or analytic sentence) • Validity can be checked by exhaustive checking of truth tables • A sentence that can be true is satisfiable

Rules of Inference • Alternative to truth-table checking • A sequence of inference rule applications leading to a desired conclusion is a logical proof • We can check inference rules using truth tables, and then use to create sound proofs • We can treat finding a proof as a search problem

Propositional Inference Rules • Modus Ponens or Implication Elimination • And Elimination • And Introduction • Unit Resolution • Resolution

Building an Agent with Propositional Logic • Propositional logic has some nice properties – Easily understood – Easily computed • Can we build a viable wumpus world agent with propositional logic???

The Problem • Propositional Logic only deals with facts. • We cannot easily represent general rules that apply to any square. • We cannot express information about squares and relate (we can’t easily keep track of which squares we have visited)

More Precisely • In propositional logic, each possible atomic fact requires a separate unique propositional symbol. • If there are n people and m locations, representing the fact that some person moved from one location to another requires nm 2 separate symbols.

First Order Logic • Predicate logic includes a richer ontology: – objects (terms) – properties (unary predicates on terms) – relations (n-ary predicates on terms) – functions (mappings from terms to other terms) • Allows more flexible and compact representation of knowledge • Move(x, y, z) for person x moved from location y to z.

Syntax for First-Order Logic Sentence  AtomicSentence | Sentence Connective Sentence | Quantifier Variable Sentence | ¬Sentence | (Sentence) AtomicSentence  Predicate(Term, Term, ...) | Term=Term Term  Function(Term,Term,...) | Constant | Variable Connective   Quanitfier  $" Constant  A | John | Car1 Variable  x | y | z |... Predicate  Brother | Owns | ... Function  father-of | plus | ...

Terms • Objects are represented by terms: – Constants: Block1, John – Function symbols: father-of, successor, plus • An n-ary function maps a tuple of n terms to another term: father-of(John), succesor(0), plus(plus(1,1),2) • Terms are simply names for objects. • Logical functions are not procedural as in programming languages. They do not need to be defined, and do not really return a value. • Functions allow for the representation of an infinite number of terms.

Predicates • Propositions are represented by a predicate applied to a tuple of terms. A predicate represents a property of or relation between terms that can be true or false: – Brother(John, Fred), Left-of(Square1, Square2) – GreaterThan(plus(1,1), plus(0,1)) • In a given interpretation, an n-ary predicate can defined as a function from tuples of n terms to {True, False} or equivalently, a set tuples that satisfy the predicate: – {<John, Fred>, <John, Tom>, <Bill, Roger>, ...}

Sentences in First-Order Logic • An atomic sentence is simply a predicate applied to a set of terms. – Owns(John,Car1) – Sold(John,Car1,Fred) • Semantics is True or False depending on the interpretation, i.e. is the predicate true of these arguments. • The standard propositional connectives (   ) can be used to construct complex sentences: – Owns(John,Car1)  Owns(Fred, Car1) – Sold(John,Car1,Fred)  ¬Owns(John, Car1) • Semantics same as in propositional logic.

Quantifiers • Allow statements about entire collections of objects • Universal quantifier: " x – Asserts that a sentence is true for all values of variable x • " x Loves(x, FOPC) • " x Whale(x)  Mammal(x) • " x (" y Dog(y)  Loves(x,y))  (" z Cat(z)  Hates(x,z)) • Existential quantifier: $ – Asserts that a sentence is true for at least one value of a variable x • $ x Loves(x, FOPC) • $ x(Cat(x)  Color(x,Black)  Owns(Mary,x)) • $ x( " y Dog(y)  Loves(x,y))  ( " z Cat(z)  Hates(x,z))

Use of Quantifiers • Universal quantification naturally uses implication: – " x Whale(x)  Mammal(x) • Says that everything in the universe is both a whale and a mammal. • Existential quantification naturally uses conjunction: – $ x Owns(Mary,x)  Cat(x) • Says either there is something in the universe that Mary does not own or there exists a cat in the universe. – " x Owns(Mary,x)  Cat(x) • Says all Mary owns is cats (i.e. everthing Mary owns is a cat). Also true if Mary owns nothing. – " x Cat(x)  Owns(Mary,x) • Says that Mary owns all the cats in the universe. Also true if there are no cats in the universe.